Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

Abstract

We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines L 1 and L 2 terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with L 1 and L 2 data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind" model selection.

A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

“…The treatment of the mixture of Gaussian and impulse noise has particularly attracted attention in recent years. One of the first successful works is presented in [36] which aims to optimize a convex functional with a total variation regularizer and a combined L 2 -L 1 data fidelity term.Recently, a successful approach for a mixture of Gaussian and impulse noise have been proposed [10,11], which is called by infimal convolution combination of data fidelities (TV-IC). In fact, this model is based on the resolution of two minimization problems.…”

“…Recently, the bi-level optimization has been considered to learn the regularization parameter in the denoising context [9,21,22,37,39,42]. In a recent work [11], the authors proposed a new procedure to learn the parameters λ 1 , λ 2 appearing in the TV-IC model (4).…”

Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal

“…If directly following the technique for BCA in the last subsection, since the subproblem for p is non-differentiable, the successive errors of Λ p cannot be controlled by the successive errors of p such that it seems impossible to guarantee the sufficient decrease of the whole iterative sequences. In order to investigate the convergence of BCA f algorithm, we adopt the Huber type TV (HTV) regularization [20,5] instead of the traditional total variation for the TV-IC model, which is Lipschitz differentiable. This change will not affect the solutions to the subproblems w.r.t.…”

Edge detection with mixed noise based on maximum a posteriori approach

“…[27]), that means there is only one term for the fidelity term Φ(u, f ). Recently, many denoising models are aimed at removing mixed noise, which degrade the spatial resolution and contrast of the restored image (see, e.g., [20,10,18]). Commonly used noises with different applications include Gaussian, Poisson, Laplace and salt & pepper (or impulsive) noises, or with some compound distributions as well.…”

“…If the noise W is Gaussian distribution with mean 0 and variance σ 2 , the Gaussian probability density for every random variable W i is (10) p Wi (w…”